In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amp...

In this work, a bidimensional continuous-time differential equations system is analyzed which is derived of Leslie-type predator-prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant, the only quadrant of interest for the sake of realism. W...

Consider a class of planar autonomous differential systems with cylindric phase space which represent generalized pendulum equations. We describe a method to construct such systems with prescribed maximum number of limit cycles which are not contractible to a point (limit cycles of the second kind). The underlying idea consists in employing Dulac-Cherkas functions. We also show how this approac...

In this paper, we study the cubic three-parameter autonomous planar system ẋ1 = k1 + k2x1 − x1 − x2, ẋ2 = k3x1 − x2, where k2, k3 > 0. Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. Results may be a...

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 10 quasi Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 10 small ...

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The linear differential two centers formed in ways, one is that a center-fold point at the origin, other origin and another unique center exists. We first explore expression of order Melnikov function. Then by using function method, give estimations numbe...

We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semiclassical app...